{"id":6539,"date":"2016-11-28T06:00:47","date_gmt":"2016-11-28T11:00:47","guid":{"rendered":"http:\/\/grockit.com\/blog\/collegeprep\/?p=2221"},"modified":"2020-09-11T20:42:31","modified_gmt":"2020-09-11T20:42:31","slug":"act-math-exponents-and-roots","status":"publish","type":"post","link":"https:\/\/wpapp.kaptest.com\/study\/act\/act-math-exponents-and-roots\/","title":{"rendered":"ACT Math: Exponents and Roots"},"content":{"rendered":"<p>&nbsp;<br \/>\n<div  style='padding-bottom:10px; ' class='av-special-heading av-special-heading-h3    avia-builder-el-0  el_before_av_heading  avia-builder-el-first  '><h3 class='av-special-heading-tag '  itemprop=\"headline\"  >Exponents<\/h3><div class='special-heading-border'><div class='special-heading-inner-border' ><\/div><\/div><\/div><br \/>\nExponents, or powers, are numbers that tell us to how many times to multiply a number by itself. For example, 2<sup>6\u00a0<\/sup>= 2 * 2 * 2 * 2 * 2 * 2. You might read this as \u201ctwo to the sixth power,\u201d and our answer would be 64. For the ACT Math, you\u2019ll really want to know how to manipulate expressions with exponents. There are many rules that dictate how we manipulate expressions with exponents, so let\u2019s get started.<br \/>\n<div  style='padding-bottom:10px; ' class='av-special-heading av-special-heading-h4  blockquote modern-quote  avia-builder-el-1  el_after_av_heading  el_before_av_heading  '><h4 class='av-special-heading-tag '  itemprop=\"headline\"  >Exponent Basics<\/h4><div class='special-heading-border'><div class='special-heading-inner-border' ><\/div><\/div><\/div><br \/>\nWhen adding algebraic expressions that have the same bases and exponents, I can add their coefficients:<br \/>\nX<sup>\u00b3<\/sup>+X<sup>\u00b3 =<\/sup>=2 X<sup>\u00b3 <\/sup><br \/>\n3X<sup>5 <\/sup>+2 X<sup>5 <\/sup>=5x<sup>5<\/sup><br \/>\nRemember that you can only add the coefficients when the base and exponent are the same. Adding and subtracting will never result in a change in exponent (e.g. 2 X<sup>\u00b3 <\/sup>+2 X<sup>\u00b3 <\/sup>does not equal 4x<sup>6 <\/sup>)<br \/>\nWhen multiplying expressions or terms that <em>have the same base,<\/em> just <em>add the exponents. <\/em><br \/>\nX<sup>\u00b3<\/sup> * X\u00b3<sup> =<\/sup>=X<sup>6 <\/sup><br \/>\nWhen dividing numbers or terms with the same base, <em>subtract <\/em>the exponents.<br \/>\nX<sup>7<\/sup> \/ X<sup>4 =<\/sup>= x\u00b3<br \/>\n<div  style='padding-bottom:10px; ' class='av-special-heading av-special-heading-h4  blockquote modern-quote  avia-builder-el-2  el_after_av_heading  el_before_av_heading  '><h4 class='av-special-heading-tag '  itemprop=\"headline\"  >Raising an Exponent to an Exponent<\/h4><div class='special-heading-border'><div class='special-heading-inner-border' ><\/div><\/div><\/div><br \/>\nWhen raising an exponent to another exponent, you must multiply the exponents. In order to raise an exponent to an exponent, you must place parentheses around the original expression and then place the exponent outside the parentheses.<br \/>\n(X<sup>\u00b3<\/sup> )\u00b3= X<sup>9 <\/sup><br \/>\nMost of the time, if students are going to mix something up about manipulating expressions with exponents, they often mix up multiplying exponents with raising an exponent to an exponent. The former involves <em>adding <\/em>exponents while the latter involves <em>multiplying <\/em>exponents.<br \/>\n<div  style='padding-bottom:10px; ' class='av-special-heading av-special-heading-h4  blockquote modern-quote  avia-builder-el-3  el_after_av_heading  el_before_av_icon_box  '><h4 class='av-special-heading-tag '  itemprop=\"headline\"  >Negative Exponents<\/h4><div class='special-heading-border'><div class='special-heading-inner-border' ><\/div><\/div><\/div><br \/>\nA number raised to a negative power is equal to \u201c1 over\u201d (a.k.a. the reciprocal of) that base raised to the opposite (positive) of that power.<br \/>\nX<sup>-\u00b3<\/sup>=1 \/ X<sup>\u00b3<\/sup><br \/>\n&nbsp;<br \/>\n<article  class=\"iconbox iconbox_left    avia-builder-el-4  el_after_av_heading  el_before_av_heading  \"  itemscope=\"itemscope\" itemtype=\"https:\/\/schema.org\/BlogPosting\" itemprop=\"blogPost\" ><div class=\"iconbox_content\"><header class=\"entry-content-header\"><div class=\"iconbox_icon heading-color\" aria-hidden='true' data-av_icon='\ue83f' data-av_iconfont='entypo-fontello'  ><\/div><h3 class='iconbox_content_title  '  itemprop=\"headline\"  >Exponent Reminders<\/h3><\/header><div class='iconbox_content_container  '  itemprop=\"text\"  ><p><strong>Zero:\u00a0<\/strong>Anything raised to the power of zero is 1<br \/>\n<strong>One:\u00a0<\/strong>Anything raised to the power of 1 is itself.<\/p>\n<\/div><\/div><footer class=\"entry-footer\"><\/footer><\/article><br \/>\n<div  style='padding-bottom:10px; ' class='av-special-heading av-special-heading-h3    avia-builder-el-5  el_after_av_icon_box  el_before_av_heading  '><h3 class='av-special-heading-tag '  itemprop=\"headline\"  >Radicals<\/h3><div class='special-heading-border'><div class='special-heading-inner-border' ><\/div><\/div><\/div><br \/>\nRoots and radicals are the inverse of exponents. If three squared (3*3) is nine, then the <em>square root <\/em>of nine (<em>what number multiplied by itself yields 9?) <\/em>is three. Note that \u201csquare roots,\u201d though the most common on the test, are not the only roots tested. The test may ask you to work with square roots, cubed roots, fourth roots, and so on. The cubed root of 8, for example, is 2, since 2*2*2 is 8.<br \/>\n<div  style='padding-bottom:10px; ' class='av-special-heading av-special-heading-h4  blockquote modern-quote  avia-builder-el-6  el_after_av_heading  el_before_av_heading  '><h4 class='av-special-heading-tag '  itemprop=\"headline\"  >Radical Basics<\/h4><div class='special-heading-border'><div class='special-heading-inner-border' ><\/div><\/div><\/div><br \/>\nYou cannot add or subtract roots. You must work out each root and then add the numbers. So, \u221a9 + \u221a16 is not \u221a25. You must convert the roots to 3 and 4, respectively, and our answer is 7.<br \/>\nYou can multiply or divide roots <em>only if <\/em>they are of the same degree (i.e. they are both square roots, cubed roots, fourth roots, etc.). So if you want to multiply two roots of the same degree, just multiply the numbers under the root sign and place that product under a new root sign with the same degree.<br \/>\n\u221a26 * \u221a2 = \u221a52<br \/>\n<div  style='padding-bottom:10px; ' class='av-special-heading av-special-heading-h4  blockquote modern-quote  avia-builder-el-7  el_after_av_heading  avia-builder-el-last  '><h4 class='av-special-heading-tag '  itemprop=\"headline\"  >Simplifying Roots<\/h4><div class='special-heading-border'><div class='special-heading-inner-border' ><\/div><\/div><\/div><br \/>\nSometimes on the test, you\u2019ll see roots in the answer choices written as the product of a number and a square root. These expressions are in simplified form. Take the \u221a52 we see above. This is not in simplified above. To simplify\u00a0 it, first find all its prime factors by drawing a factor tree<br \/>\n52<br \/>\n\/ \u00a0 \\<br \/>\n<strong>2 \u00a0<\/strong>26<br \/>\n\/ \u00a0 \u00a0 \\<br \/>\n13\u00a0\u00a0\u00a0 <strong>2<\/strong><br \/>\no\u00a0\u00a0 Notice that the prime factors of 52 are 2, 2, and 13. We can write \u221a52 as a product of 2 and \u221a13. When simplifying radicals, we want to identify the <em>pairs <\/em>of prime factors; in this case, we have two 2s, so while we didn\u2019t notice it in the first place, we can write \u221a52 as \u221a13*4, which we can write as \u221a13 * \u221a4, which simplifies to 2 \u221a13.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>&nbsp; Exponents, or powers, are numbers that tell us to how many times to multiply a number by itself. For example, 26\u00a0= 2 * 2 * 2 * 2 * 2 * 2. You might read this as \u201ctwo to the sixth power,\u201d and our answer would be 64. For the ACT Math, you\u2019ll really [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27027,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[58],"tags":[792],"_links":{"self":[{"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/posts\/6539"}],"collection":[{"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/comments?post=6539"}],"version-history":[{"count":3,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/posts\/6539\/revisions"}],"predecessor-version":[{"id":36119,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/posts\/6539\/revisions\/36119"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/media\/27027"}],"wp:attachment":[{"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/media?parent=6539"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/categories?post=6539"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/tags?post=6539"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}